Editing Worldsheet (section)

=== Bosonic string ===

We begin with the classical formulation of the bosonic string. 

First fix a <math>d</math>-dimensional [[flat (geometry)|flat]] [[spacetime]] (<math>d</math>-dimensional [[Minkowski space]]), <math>M</math>, which serves as the [[ambient space]] for the string.

A '''world-sheet''' <math>\Sigma</math> is then an [[embedding|embedded]] [[surface (topology)|surface]], that is, an embedded 2-manifold <math>\Sigma \hookrightarrow M</math>, such that the [[induced metric]] has signature <math>(-,+)</math> everywhere. Consequently it is possible to locally define coordinates <math>(\tau,\sigma)</math> where <math>\tau</math> is [[time-like]] while <math>\sigma</math> is [[space-like]].

Strings are further classified into open and closed. The topology of the worldsheet of an open string is <math>\mathbb{R}\times I</math>, where <math>I := [0,1]</math>, a closed interval, and admits a global coordinate chart <math>(\tau, \sigma)</math> with <math>-\infty < \tau < \infty</math> and <math>0 \leq \sigma \leq 1</math>. 

Meanwhile the topology of the worldsheet of a closed string<ref name=tong>{{cite web |url=http://www.damtp.cam.ac.uk/user/tong/string.html |title=Lectures on String Theory |last=Tong |first=David |website=Lectures on Theoretical Physics |access-date=August 14, 2022}}</ref> is <math>\mathbb{R}\times S^1</math>, and admits 'coordinates' <math>(\tau, \sigma)</math> with <math>-\infty < \tau < \infty</math> and <math>\sigma \in \mathbb{R}/2\pi\mathbb{Z}</math>. That is, <math>\sigma</math> is a periodic coordinate with the identification <math>\sigma \sim \sigma + 2\pi</math>. The redundant description (using quotients) can be removed by choosing a representative <math>0 \leq \sigma < 2\pi</math>.

==== World-sheet metric ====
In order to define the [[Polyakov action]], the world-sheet is equipped with a '''world-sheet metric'''<ref name="Polchinski">{{cite book|last1=Polchinski|first1=Joseph|year=1998|title=String Theory, Volume 1: Introduction to the Bosonic string}}</ref> <math>\mathbf{g}</math>, which also has signature <math>(-, +)</math> but is independent of the induced metric.

Since [[Weyl transformation]]s are considered a redundancy of the metric structure, the world-sheet is instead considered to be equipped with a [[conformal class]] of metrics <math>[\mathbf{g}]</math>. Then <math>(\Sigma, [\mathbf{g}])</math> defines the data of a [[conformal manifold]] with signature <math>(-, +)</math>.